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An option's price splits into two parts that are worth separating before any model is involved.

Intrinsic value is what the contract is worth if exercised right now, floored at zero:

$\text{call} = \max(S - K,\, 0), \qquad \text{put} = \max(K - S,\, 0)$

where $S$ is the underlying price and $K$ the strike.

Extrinsic value (time value) is everything left over:

$\text{extrinsic} = \text{premium} - \text{intrinsic}$

It is never negative, and it decays to zero at expiry.

Payoff and breakeven for a long position, per share, ignoring commissions:

$\text{call profit} = \max(S_T - K, 0) - \text{premium}, \qquad \text{breakeven} = K + \text{premium}$
$\text{put profit} = \max(K - S_T, 0) - \text{premium}, \qquad \text{breakeven} = K - \text{premium}$

A US equity contract normally covers 100 shares, so multiply per-share figures by 100.

This is a math tool. It evaluates the formulas on inputs you supply — it makes no recommendation about any trade.

For a European option on a non-dividend-paying underlying:

$C = S\,N(d_1) - Ke^{-rT}N(d_2), \qquad P = Ke^{-rT}N(-d_2) - S\,N(-d_1)$

$d_1 = \frac{\ln(S/K) + \left(r + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$

• $S$ — underlying price, $K$ — strike
• $r$ — risk-free rate, continuously compounded, as a decimal
• $\sigma$ — annualized volatility as a decimal
• $T$ — time to expiry in years (30 days $\approx 30/365 \approx 0.0822$)
• $N(\cdot)$ — the standard normal CDF

Put-call parity

$C - P = S - Ke^{-rT}$

A free consistency check: price the call, then get the put from parity instead of running the second formula.

Delta

$\Delta_{\text{call}} = N(d_1), \qquad \Delta_{\text{put}} = N(d_1) - 1$

Delta is the sensitivity of the price to a small move in $S$, and is often read loosely as a rough gauge of finishing in the money.

Black-Scholes assumes constant volatility, no dividends and European exercise — none of which hold exactly.

• $T$ in days: the formula needs years. Divide the days by 365 (or 252 if you are working in trading days consistently with $\sigma$).
• Volatility as a percentage: $\sigma = 0.20$, not $20$. This error inflates prices absurdly.
• Forgetting the $\sigma^2/2$ term in $d_1$: it is not a typo in the textbook; dropping it biases every price.
• Discounting the wrong leg: only the strike is discounted, as $Ke^{-rT}$. The spot price $S$ is already a present value.
• Confusing $N(d)$ with the density: $N$ is the cumulative standard normal, the same $\Phi(z)$ used for z-scores.
• Negative extrinsic value: if your computed extrinsic value is below zero, the intrinsic value or the quote is wrong — the two cannot cross.
• Applying European Black-Scholes to American options: early exercise can add value, particularly for puts and around dividends.
• Per-share versus per-contract: an \4.80$per-share profit is$$480$ on a 100-share contract.